
 A divisor \(D\in \operatorname{Div}(G)\)  is called admissible if \(D(v)=1\)  for any \(v\in V_{\operatorname{exc}}(G)\) .

 Let \(\phi \in V_{g,n}\)  be a universal stability condition.

For a subset \(S\subseteq V(G)\)  we write \(D(S):=\sum _{v\in S} D(v)\)  as well as \(\phi _G(S):=\sum _{v\in S} \phi _G(v)\) . A divisor \(D\in \operatorname{Div}(G)\)  is called \(\phi \) -semistable if \(D\)  is admissible of degree \(\deg D\)  equal to \(\vert \phi \vert \)  and the following inequalities
hold for any subset \(S\subseteq V(G)\) :
\(\phi _G(S)-\frac{\big |E(S,S^c)\big |}{2}\le D(S)\le \phi _G(S)+\frac{\big |E(S,S^c)\big |}{2},\) 
where \(E(S,S^c)\)  is the set of edges joining a vertex in \(S\)  with a vertex in the complementary subset \(S^c\) .

A divisor \(D\in \operatorname{Div}(G)\)  is called \(\phi \) -stable if \(D\)  is \(\phi \) -semistable and both the inequalities (REF ) are strict unless \(S\)  or \(S^c\)  is a union (possibly empty) of exceptional vertices.


 A universal stability condition \(\phi \in V_{g,n}\)  is called general This is called non-degenrate in [1]}. We prefer to call it general according to the terminology used in [2]}, [3]}, [4]}, [5]}, [6]}. Note that in the first two papers, the term non-degenerate is used for a slightly weaker condition. if for any quasi-stable graph of type \((g,n)\)  we have that for any subset \(S\subseteq V(G)\)
\( \phi _G(S)+\frac{\big |E(S,S^c)\big |}{2}\in \mathbb {Z}\Rightarrow S \text{ or } S^c \text{ is a union of exceptional vertices.} \) 


